Mathematical Research Letters

Volume 28 (2021)

Number 2

qKZ/tRS duality via quantum $K$-theoretic counts

Pages: 435 – 470



Peter Koroteev (Department of Mathematics, University of California, Berkeley, Cal., U.S.A.)

Anton M. Zeitlin (Department of Mathematics, Louisiana State University, Baton Rouge, La., U.S.A.; and IPME RAS, St. Petersburg, Russia)


We show that normalized quantum $K$-theoretic vertex functions for cotangent bundles of partial flag varieties are the eigenfunctions of quantum trigonometric Ruijsenaars–Schneider (tRS) Hamiltonians. Using recently observed relations between quantum Knizhnik–Zamolodchikov (qKZ) equations and tRS integrable system we derive a nontrivial identity for vertex functions with relative insertions.

The work of A.M. Zeitlin is partially supported by Simons Collaboration Grant, Award ID: 578501.

P.K. acknowledges support of IHÉS and funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (QUASIFT grant agreement 677368).

Received 24 February 2019

Accepted 26 October 2019

Published 13 May 2021